The existence of partitioned generalized balanced tournament designs with block size 3
Let \(V\) be a set with \(kn\) elements. If the blocks of a \(\text{BIBD}(kn,k,k-1)\) defined on \(V\) can be arranged into an \(n\times(nk-1)\) array such that (i) each element in \(V\) occurs precisely in one cell of each column, and (ii) every element in \(V\) is contained in at most \(k\) cells of each row, then the configuration so obtained is called a generalized balanced tournament design, denoted by \(\text{GBTD}(n,k)\). Suppose, further, we can partition the columns of GBTD into \((k+1)\) sets \(B_1,B_2,\dots,B_{k+1}\) where \(|B_i|= n\) for \(i=1,2,3,\dots,k-2\), \(|B_i|= n-1\) for \(i=k-1, k\) and \(|B_i|=1\) for \(i=k+1\) such that (i) each element of \(V\) appears exactly once in each row and column of \(B_i\) \((i=1,2,\dots,k-2)\), and (ii) every element of \(V\) occurs exactly once in each row and column of \(B_i\cup B_{i+1}\) for \(i=k-1, k\), then GBTD is called partitioned, denoted by \(\text{PGBTD}(n,k)\). The author proves the existence of \(\text{PGBTD}(n,k)\) for \(k=3\) with the possible exceptions of 165 values of \(n\). These results are then used to establish the existence of a class of Kirkman squares in diagonal form. Finally, an appendix containing tables is given which describe the constructions for \(\text{PGBTD}(n',3)\) with \(461\leq n'\leq 1745\) and \(n'\equiv i\pmod 9\) for \(i=2,3,\dots,8\). A computer program was used to generate these tables.
scientific article; zbMATH DE number 1002297